The PBW filtration and convex polytopes in type B

We study the PBW filtration on irreducible finite-dimensional representations for the Lie algebra of type ${\mathtt{B}}_n$. We prove in various cases, including all multiples of the adjoint representation and all irreducible finite-dimensional representations for ${\mathtt{B}}_3$, that there exists a normal polytope such that the lattice points of this polytope parametrize a basis of the corresponding associated graded space. As a consequence we obtain several classes of examples for favourable modules and graded combinatorial character formulas.